properties of subtraction
What are the properties of subtraction?
Answer: Subtraction, denoted usually as “-”, is one of the four elementary operations in arithmetic. Unlike addition, multiplication, and division, subtraction does not possess as many properties due to its non-commutative and non-associative nature. Below, we examine the primary properties and characteristics associated with subtraction:
1. Non-Commutativity:
Subtraction is not commutative. This means that changing the order of the numbers in a subtraction operation changes the result.
a - b \neq b - a
For example:
5 - 3 \neq 3 - 5
2 \neq -2
2. Non-Associativity:
Subtraction is not associative. This means that when subtracting three or more numbers, the way in which pairs of numbers are grouped affects the result.
(a - b) - c \neq a - (b - c)
For example:
(10 - 5) - 2 \neq 10 - (5 - 2)
5 - 2 \neq 10 - 3
3 \neq 7
3. Identity Property:
The identity element for subtraction is zero. If you subtract zero from any number, the result is the number itself.
$$ a - 0 = a $$
For example:
8 - 0 = 8
4. Subtractive Inverse:
Every number has a subtractive inverse, which is the same number itself. If a number is subtracted by itself, the result is zero.
a - a = 0
For example:
7 - 7 = 0
5. Order Property:
Subtraction follows the order property where the larger number subtracted by the smaller number gives a positive value, and the smaller number subtracted by the larger number gives a negative value:
\text{If} \ a > b \ \text{then} \ a - b > 0
\text{If} \ a < b \ \text{then} \ a - b < 0
For example:
\text{If} \ 9 > 4 \ \text{then} \ 9 - 4 = 5 > 0
\text{If} \ 3 < 7 \ \text{then} \ 3 - 7 = -4 < 0
6. Distributive Property over Addition (of subtraction):
For any three numbers ( a ), ( b ), and ( c ):
a - (b + c) = (a - b) - c
For example:
10 - (2 + 3) = (10 - 2) - 3
10 - 5 = 8 - 3
5 = 5
7. Existence of Negative Results:
A unique trait of subtraction is it can yield negative results if the subtrahend (number being subtracted) is larger than the minuend (number from which a value is subtracted).
For example:
3 - 8 = -5
Example Exercises:
-
Non-Communtativity Example:
9 - 4 = 5
4 - 9 = -5 -
Non-Associativity Example:
(15 - 5) - 3 = 10 - 3 = 7
15 - (5 - 3) = 15 - 2 = 13 -
Identity Property Example:
12 - 0 = 12 -
Subtractive Inverse Example:
6 - 6 = 0
Understanding these properties is integral to mastering subtraction. They highlight how subtraction fundamentally differs from other arithmetic operations and frame our approach to arithmetic manipulations involving subtraction.
By exploring and practicing these properties, students can better grasp the structure and nuances of subtraction, aiding in their overall arithmetic proficiency.